Two-dimensional symmetry-protected topological orders and their protected gapless edge excitations
Topological insulators in free fermion systems have been well characterized and classified. However, it is not clear in strongly interacting boson or fermion systems what symmetry-protected topological orders exist. In this paper, we present a model in a two-dimensional (2D) interacting spin system...
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| Format: | Article |
| Language: | en_US |
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American Physical Society
2012
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| Online Access: | http://hdl.handle.net/1721.1/72029 https://orcid.org/0000-0002-5874-581X |
| _version_ | 1826210228489682944 |
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| author | Chen, Xie Liu, Zheng-Xin Wen, Xiao-Gang |
| author2 | Massachusetts Institute of Technology. Department of Physics |
| author_facet | Massachusetts Institute of Technology. Department of Physics Chen, Xie Liu, Zheng-Xin Wen, Xiao-Gang |
| author_sort | Chen, Xie |
| collection | MIT |
| description | Topological insulators in free fermion systems have been well characterized and classified. However, it is not clear in strongly interacting boson or fermion systems what symmetry-protected topological orders exist. In this paper, we present a model in a two-dimensional (2D) interacting spin system with nontrivial onsite Z[subscript 2] symmetry-protected topological order. The order is nontrivial because we can prove that the one-dimensional (1D) system on the boundary must be gapless if the symmetry is not broken, which generalizes the gaplessness of Wess-Zumino-Witten model for Lie symmetry groups to any discrete symmetry groups. The construction of this model is related to a nontrivial 3-cocycle of the Z[subscript 2] group and can be generalized to any symmetry group. It potentially leads to a complete classification of symmetry-protected topological orders in interacting boson and fermion systems of any dimension. Specifically, this exactly solvable model has a unique gapped ground state on any closed manifold and gapless excitations on the boundary if Z[subscript 2] symmetry is not broken. We prove the latter by developing the tool of a matrix product unitary operator to study the nonlocal symmetry transformation on the boundary and reveal the nontrivial 3-cocycle structure of this transformation. Similar ideas are used to construct a 2D fermionic model with onsite Z[subscript 2] symmetry-protected topological order. |
| first_indexed | 2024-09-23T14:46:29Z |
| format | Article |
| id | mit-1721.1/72029 |
| institution | Massachusetts Institute of Technology |
| language | en_US |
| last_indexed | 2024-09-23T14:46:29Z |
| publishDate | 2012 |
| publisher | American Physical Society |
| record_format | dspace |
| spelling | mit-1721.1/720292022-10-01T22:21:32Z Two-dimensional symmetry-protected topological orders and their protected gapless edge excitations Chen, Xie Liu, Zheng-Xin Wen, Xiao-Gang Massachusetts Institute of Technology. Department of Physics Wen, Xiao-Gang Chen, Xie Liu, Zheng-Xin Wen, Xiao-Gang Topological insulators in free fermion systems have been well characterized and classified. However, it is not clear in strongly interacting boson or fermion systems what symmetry-protected topological orders exist. In this paper, we present a model in a two-dimensional (2D) interacting spin system with nontrivial onsite Z[subscript 2] symmetry-protected topological order. The order is nontrivial because we can prove that the one-dimensional (1D) system on the boundary must be gapless if the symmetry is not broken, which generalizes the gaplessness of Wess-Zumino-Witten model for Lie symmetry groups to any discrete symmetry groups. The construction of this model is related to a nontrivial 3-cocycle of the Z[subscript 2] group and can be generalized to any symmetry group. It potentially leads to a complete classification of symmetry-protected topological orders in interacting boson and fermion systems of any dimension. Specifically, this exactly solvable model has a unique gapped ground state on any closed manifold and gapless excitations on the boundary if Z[subscript 2] symmetry is not broken. We prove the latter by developing the tool of a matrix product unitary operator to study the nonlocal symmetry transformation on the boundary and reveal the nontrivial 3-cocycle structure of this transformation. Similar ideas are used to construct a 2D fermionic model with onsite Z[subscript 2] symmetry-protected topological order. National Science Foundation (U.S.) (grant no. NSFC 11074140) 2012-08-08T15:24:56Z 2012-08-08T15:24:56Z 2011-12 2011-10 Article http://purl.org/eprint/type/JournalArticle 1098-0121 1550-235X http://hdl.handle.net/1721.1/72029 Chen, Xie, Zheng-Xin Liu, and Xiao-Gang Wen. “Two-dimensional Symmetry-protected Topological Orders and Their Protected Gapless Edge Excitations.” Physical Review B 84.23 (2011): 235141-1-235141-13. ©2011 American Physical Society https://orcid.org/0000-0002-5874-581X en_US http://dx.doi.org/10.1103/PhysRevB.84.235141 Physical Review B Article is made available in accordance with the publisher's policy and may be subject to US copyright law. Please refer to the publisher's site for terms of use. application/pdf American Physical Society APS |
| spellingShingle | Chen, Xie Liu, Zheng-Xin Wen, Xiao-Gang Two-dimensional symmetry-protected topological orders and their protected gapless edge excitations |
| title | Two-dimensional symmetry-protected topological orders and their protected gapless edge excitations |
| title_full | Two-dimensional symmetry-protected topological orders and their protected gapless edge excitations |
| title_fullStr | Two-dimensional symmetry-protected topological orders and their protected gapless edge excitations |
| title_full_unstemmed | Two-dimensional symmetry-protected topological orders and their protected gapless edge excitations |
| title_short | Two-dimensional symmetry-protected topological orders and their protected gapless edge excitations |
| title_sort | two dimensional symmetry protected topological orders and their protected gapless edge excitations |
| url | http://hdl.handle.net/1721.1/72029 https://orcid.org/0000-0002-5874-581X |
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